Non-diagonal anisotropic quantum Hall states

We introduce a family of Abelian quantum Hall states termed the non-diagonal states, arising at filling ν = p/2q for bosonic systems and ν = p/(p + 2q) for fermionic systems, with p and q being two coprime integers. Non-diagonal states are constructed in a coupled wire model, which shows an intimate relation to the non-diagonal conformal field theory and features quasiparticles with constrained motion. From the quasiparticle braiding statistics, we establish the non-diagonal state as a symmetry-enriched topological order that mixes a Laughlin state with a Z_p toric code. Two symmetries are relevant: the U(1) charge symmetry and the Z translation symmetry of the wire model. In particular, translation symmetry distinguishes non-diagonal states from Laughlin states, in a way similar to how it distinguishes weak topological insulators from trivial band insulators. Moreover, translation symmetry in non-diagonal states is associated to the e ↔ m anyonic symmetry in Z_p toric code, implying the role of dislocations as two-fold twist-defects. Thus, our microscopic model also provides a route of realizing an isotropic non-Abelian states through the melting of wire model.